2.3 Atmospheric Equilibrium
When water is in contact with the atmosphere (at a constant pressure, temperature, and humidity) for sufficient time, an equilibrium concentration is reached that can be calculated using Henry’s Law. While the Henry Coefficient is a function of both temperature and salinity, it is not a function of the total atmospheric pressure. However, the equilibrium concentration of air in water is a function of the total gas pressure and hence is a function of elevation. Equation (4) is a modification of Henry’s Law that gives the concentration of gas [latex]x_{i}[/latex] in air‑saturated water (ASW).
| [latex]C_{i,eq} = \frac{\left( P - e(T) \right)x_{i}}{K_{w,i}(T,S)}[/latex] | (4) |
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where:
| [latex]C_{i,eq}[/latex] | = | equilibrium concentration of gas i in air‑saturated water (L3M-1, typically in ccSTP/g) |
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| P | = | atmospheric pressure (ML-1T-2, typically in atm) |
| e(T) | = | vapor pressure of water at temperature T (ML-1T-2, typically in atm) |
| [latex]K_{w,i}(T,S)[/latex] | = | Henry Coefficient of gas i at temperature T and salinity S (M2L-4T-2, typically in g‑atm/ccSTP) |
| [latex]x_{i}[/latex] | = | fraction of gas i (mixing ratio) in the dry atmosphere (dimensionless) |
The atmospheric pressure (P) is a function of elevation (Z) and can be closely approximated by Equation (5).
| [latex]P = P_{0}\exp\left( - \frac{Z}{Z_{s}} \right)[/latex] | (5) |
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where:
| [latex]P_{\ }[/latex] | = | atmospheric pressure (ML-1T-2, typically in atm) |
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| [latex]P_{0}[/latex] | = | pressure at elevation (Z) = 0 (ML-1T-2, defined as 1 atm) |
| [latex]Z[/latex] | = | Elevation (L) |
| [latex]Z_{s}[/latex] | = | scaling factor (L) |
Using a value for [latex]Z_{s}[/latex] of 8,300 m this expression is a very good approximation for [latex]Z[/latex] < 1,800 m. The vapor pressure of water (e(T)) in Equation (4) is used because the composition of the atmosphere is typically given without water vapor since the vapor pressure of water is highly variable. For groundwater recharge at the water table, the relative humidity of the soil atmosphere is usually assumed to be 100 percent and then the partial pressure of water vapor depends only on temperature. There are many empirical formulas for computing e as a function of temperature. The simple Antoine equation (Antoine, 1888) states the value of e as shown in Equation (6).
| [latex]e = 10^{A\ - \ \frac{B}{C + T}}[/latex] | (6) |
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where:
| [latex]e_{\ }[/latex] | = | vapor pressure of water in mmHg |
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| A | = | 8.07131 |
| B | = | 1,730.63 |
| C | = | 233.426 |
| T | = | temperature in °C |
In this calculation, e is accurate to better than one percent between 0 °C and 100 °C.